JEE MAIN - Mathematics (2021 - 26th August Morning Shift - No. 8)
If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point ($$-$$30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is :
5
7
5$${\sqrt 3 }$$
3$${\sqrt 5 }$$
Explanation
Equation of tangent to y2 = 30 x
y = mx + $${{30} \over {4m}}$$
Pass thru ($$-$$30, 0) : a = $$-$$30m + $${{30} \over {4m}}$$ $$\Rightarrow$$ m2 = 1/4
$$\Rightarrow$$ m = $${1 \over 2}$$ or m = $$-$$$${1 \over 2}$$
At m = $${1 \over 2}$$ : y = $${x \over 2}$$ + 15 $$\Rightarrow$$ x $$-$$ 2y + 30 = 0
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lAB = 2$$\sqrt {{R^2} - {P^2}} $$ = 2$$\sqrt {{{225} \over 4} - {{225} \over 5}} $$
$$\Rightarrow$$ lAB = 30 . $$\sqrt {{1 \over {20}}} $$ = $${{{15} \over {\sqrt 5 }}}$$ = 3$${\sqrt 5 }$$
y = mx + $${{30} \over {4m}}$$
Pass thru ($$-$$30, 0) : a = $$-$$30m + $${{30} \over {4m}}$$ $$\Rightarrow$$ m2 = 1/4
$$\Rightarrow$$ m = $${1 \over 2}$$ or m = $$-$$$${1 \over 2}$$
At m = $${1 \over 2}$$ : y = $${x \over 2}$$ + 15 $$\Rightarrow$$ x $$-$$ 2y + 30 = 0
lAB = 2$$\sqrt {{R^2} - {P^2}} $$ = 2$$\sqrt {{{225} \over 4} - {{225} \over 5}} $$
$$\Rightarrow$$ lAB = 30 . $$\sqrt {{1 \over {20}}} $$ = $${{{15} \over {\sqrt 5 }}}$$ = 3$${\sqrt 5 }$$
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