{"id":679485,"date":"2023-09-03T03:34:55","date_gmt":"2023-09-02T22:04:55","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/if-%ce%b1%ce%b2-are-the-two-zeros-of-the-polynomial-fyy2-8ya-and-%ce%b12%ce%b2240-find-the-value-of-a\/"},"modified":"2025-06-20T13:28:04","modified_gmt":"2025-06-20T07:58:04","slug":"if-%ce%b1%ce%b2-are-the-two-zeros-of-the-polynomial-fyy2-8ya-and-%ce%b12%ce%b2240-find-the-value-of-a","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/if--are-the-two-zeros-of-the-polynomial-fyy2-8ya-and-2\/","title":{"rendered":"If \u03b1,\u03b2 are the two zeros of the polynomial fy=y2-8y+a and \u03b12+\u03b22=40. Find the value of a."},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"If \u03b1,\u03b2 are the two zeros of the polynomial fy=y2-8y+a and \u03b12+\u03b22=40. Find the value of a.","custom_permalink":"question\/mathematics\/if--are-the-two-zeros-of-the-polynomial-fyy2-8ya-and-2\/"},"categories":[],"acf":{"question":"

If <\/span>\u03b1<\/mi>,<\/mo>\u03b2<\/mi><\/math> are the two zeros of the polynomial <\/span>f<\/mi>y<\/mi><\/mrow><\/mfenced>=<\/mo>y<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>8<\/mn>y<\/mi>+<\/mo>a<\/mi><\/math> and <\/span>\u03b1<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>+<\/mo>\u03b2<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>=<\/mo>40<\/mn><\/math>. Find the value of a.<\/span><\/p>

<\/p>

<\/p><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\"><\/spanstyle=\"font-family:'timesnewroman';-aw-import:ignore\"><\/pstyle=\"margin-top:0pt;margin-bottom:0pt;line-height:normal;font-size:12pt\">","options":[{"option":" <\/span>a<\/mi>=<\/mo>17<\/mn><\/math>","correct":false},{"option":" <\/span>a<\/mi>=<\/mo>12<\/mn><\/math>","correct":true},{"option":" <\/span>a<\/mi>=<\/mo>16<\/mn><\/math>","correct":false},{"option":" <\/span>a<\/mi>=<\/mo>42<\/mn><\/math> <\/span>","correct":false}],"solution":"Given that, <\/span>f<\/mi>y<\/mi><\/mrow><\/mfenced>=<\/mo>y<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>8<\/mn>y<\/mi>+<\/mo>a<\/mi><\/math> and <\/span>\u03b1<\/mi><\/math> and <\/span>\u03b2<\/mi><\/math> are the zeroes of the polynomial f(x), <\/span>\u03b1<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>+<\/mo>\u03b2<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>=<\/mo>40<\/mn><\/math>.<\/span>
Comparing the given equation with <\/span>a<\/mi>y<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>+<\/mo>by<\/mi>+<\/mo>c<\/mi><\/math>,<\/span>
a<\/mi>=<\/mo>1<\/mn><\/math>, <\/span>
b<\/mi>=<\/mo>-<\/mo>8<\/mn><\/math>, <\/span>
c<\/mi>=<\/mo>a<\/mi><\/math> <\/span>Now, sum of roots is,<\/span>
\u03b1<\/mi>+<\/mo>\u03b2<\/mi>=<\/mo>-<\/mo>b<\/mi><\/mrow>a<\/mi><\/mrow><\/mfrac><\/math> <\/span>\u21d2<\/mo>\u03b1<\/mi>+<\/mo>\u03b2<\/mi>=<\/mo>-<\/mo>-<\/mo>8<\/mn><\/mrow><\/mfenced><\/mrow>1<\/mn><\/mrow><\/mfrac><\/math> <\/span>\u21d2<\/mo>\u03b1<\/mi>+<\/mo>\u03b2<\/mi>=<\/mo>8<\/mn> <\/mi><\/math> <\/span>And, product of roots is,<\/span>
\u03b1\u03b2<\/mi>=<\/mo>c<\/mi><\/mrow>a<\/mi><\/mrow><\/mfrac><\/math> <\/span>\u21d2<\/mo>\u03b1\u03b2<\/mi>=<\/mo>a<\/mi><\/mrow>1<\/mn><\/mrow><\/mfrac><\/math> <\/span>\u21d2<\/mo>\u03b1\u03b2<\/mi>=<\/mo>a<\/mi><\/math> <\/span>Now, using the identity,<\/span>
\u03b1<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>+<\/mo>\u03b2<\/mi><\/mrow>2<\/mn><\/mrow><\/msup>=<\/mo>(<\/mo>\u03b1<\/mi>+<\/mo>\u03b2<\/mi>)<\/mo><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>2<\/mn>\u03b1\u03b2<\/mi><\/math> <\/span>Substituting the values,<\/span>
40<\/mn>=<\/mo> <\/mi>8<\/mn><\/mrow>2<\/mn><\/mrow><\/msup>-<\/mo>2<\/mn>a<\/mi><\/math> <\/span>2a = 24<\/span>
Thus, the value of a = 12.<\/span>
Hence, option 2 is correct.<\/span>
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