The focal chord of the parabola (y$$−$$2)2=16(x$$−$$1) is a tangent to the circle $$x2+y2$$−$$14x$$−$$4y+51=0$$, then slope of the focal chord can be

Question

The focal chord of the parabola (y$$−$$2)2=16(x$$−$$1) is a tangent to the circle $$x2+y2$$−$$14x$$−$$4y+51=0$$, then slope of the focal chord can be

Infinity Learn · Accepted Answer

Given parabola is (y$$−$$2)2=16(x$$−$$1) Here vertex $$=(h$$,$$k)=(1$$,$$2)4a=16$$⇒$$a=4$$ Focus $$=(h+a$$,$$k)=(1+4$$,$$2)=(5$$,$$2)$$ The equation of focal chord is y−$$2=mx$$−$$5$$⇒mx−$$y+2$$−$$5m=0$$………...(1) Given circle equation is $$x2+y2$$−$$14x$$−$$4y+51=0$$………...(2) Hence centre (7$$,$$2) And radius, $$r=49+4$$−$$51=2Since$$ equation (1) is a tangent of circle equation (2) thenr $$=$$ $$d2=$$|$$7m$$−$$2+2$$−$$5m$$|$$m2+1($$∵$$d=$$ perpendiculardistance from centre to equation (1))⇒$$2m2+1=$$|$$2m$$| Squaring on both sides ⇒$$2m2=m2+1$$⇒$$m=$$±$$1$$∴ Slope of focal chord is $$1$$

$The focal chord of the parabola (y - 2)^{2} = 16 (x - 1) is a tangent to the circle$
$x^{2} + y^{2} - 14 x - 4 y + 51 = 0, then slope of the focal chord can be$

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The focal chord of the parabola (y−2)2=16(x−1) is a tangent to the circle x2+y2−14x−4y+51=0, then slope of the focal chord can be

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$The focal chord of the parabola (y - 2)^{2} = 16 (x - 1) is a tangent to the circle$
$x^{2} + y^{2} - 14 x - 4 y + 51 = 0, then slope of the focal chord can be$