{"id":299859,"date":"2022-09-10T21:33:50","date_gmt":"2022-09-10T16:03:50","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/if-lines-xyz-and-xy-2z-3-and-third-line-passing-through-111-form-a-triangle-of-area-6-units-then-point-of-intersection-of-third-line-with-second-line\/"},"modified":"2022-09-10T21:33:50","modified_gmt":"2022-09-10T16:03:50","slug":"if-lines-xyz-and-xy-2z-3-and-third-line-passing-through-111-form-a-triangle-of-area-6-units-then-point-of-intersection-of-third-line-with-second-line","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/if-linesxyzandxy2z3and-third-line-passing-through\/","title":{"rendered":"If lines\u00a0x=y=z\u00a0and\u00a0x=y\/2=z\/3\u00a0and third line passing through\u00a0(1,1,1)\u00a0form a triangle of area \u00a06\u00a0units, then point of intersection of third line with second line will be"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"If lines\u00a0x=y=z\u00a0and\u00a0x=y\/2=z\/3\u00a0and third line passing through\u00a0(1,1,1)\u00a0form a triangle of area \u00a06\u00a0units, then point of intersection of third line with second line will be","custom_permalink":"question\/mathematics\/if-linesxyzandxy2z3and-third-line-passing-through\/"},"categories":[],"acf":{"question":"

If lines x<\/mi>=<\/mo>y<\/mi>=<\/mo>z<\/mi><\/math> and x<\/mi>=<\/mo>y<\/mi>\/<\/mo>2<\/mn>=<\/mo>z<\/mi>\/<\/mo>3<\/mn><\/math> and third line passing through (<\/mo>1<\/mn>,<\/mo>1<\/mn>,<\/mo>1<\/mn>)<\/mo><\/math> form a triangle of area  6<\/mn><\/msqrt><\/math> units, then point of intersection of third line with second line will be<\/p>","options":[{"option":"

(<\/mo>1<\/mn>,<\/mo>2<\/mn>,<\/mo>3<\/mn>)<\/mo><\/math><\/p>","correct":false},{"option":"

(<\/mo>2<\/mn>,<\/mo>4<\/mn>,<\/mo>6<\/mn>)<\/mo><\/math><\/p>","correct":true},{"option":"

4<\/mn><\/mstyle>3<\/mn><\/mstyle><\/mfrac>,<\/mo>8<\/mn><\/mstyle>3<\/mn><\/mstyle><\/mfrac>,<\/mo>12<\/mn><\/mstyle>3<\/mn><\/mstyle><\/mfrac><\/mrow><\/mfenced><\/math><\/p>","correct":false},{"option":"

(<\/mo>2<\/mn>,<\/mo>1<\/mn>,<\/mo>3<\/mn>)<\/mo><\/math><\/p>","correct":false}],"solution":"

x = y = z--- (1),    x<\/mi>=<\/mo>y<\/mi>2<\/mn><\/mfrac>=<\/mo>z<\/mi>3<\/mn><\/mfrac><\/math> --- (2)
Clearly point of intersection of (1) and (2) is (0,0,0)  
D.r\u2019s of (1) are (1, 1, 1)
D.r\u2019s of (2) are (1, 2, 3)
Let \u03b8<\/mi><\/math> be the angle between (1) and (2)
cos<\/mi>\u03b8<\/mi>=<\/mo>6<\/mn>42<\/mn><\/msqrt><\/mfrac>,<\/mo>sin<\/mi>\u03b8<\/mi>=<\/mo>6<\/mn><\/msqrt>42<\/mn><\/msqrt><\/mfrac><\/math>
Let any point on second line be (<\/mo>\u03bb<\/mi>,<\/mo>2<\/mn>\u03bb<\/mi>,<\/mo>3<\/mn>\u03bb<\/mi>)<\/mo><\/math>
Third line passing through (1, 1, 1)
(1, 1, 1) lies on (1)
A = (1, 1, 1)<\/p>

A<\/mi>r<\/mi>e<\/mi>a<\/mi> <\/mo>o<\/mi>f<\/mi> <\/mo>\u0394<\/mi>O<\/mi>A<\/mi>B<\/mi>=<\/mo>1<\/mn>2<\/mn><\/mfrac>(<\/mo>O<\/mi>A<\/mi>)<\/mo><\/mrow>O<\/mi>B<\/mi>sin<\/mi>\u03b8<\/mi><\/math><\/p>

=<\/mo>1<\/mn>2<\/mn><\/mfrac>3<\/mn><\/msqrt>\u03bb<\/mi>14<\/mn><\/msqrt>\u00d7<\/mo>6<\/mn><\/msqrt>42<\/mn><\/msqrt><\/mfrac>=<\/mo>6<\/mn><\/msqrt><\/math>\u21d2<\/mo>\u03bb<\/mi>=<\/mo>2<\/mn><\/math><\/p>

So B is (<\/mo>2<\/mn>,<\/mo>4<\/mn>,<\/mo>6<\/mn>)<\/mo><\/math><\/p>
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