"Let y = mx + c, m 0 be the focal chord of y2 = $-$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________."

Question

"Let y = mx + c, m > 0 be the focal chord of y2 = $-$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________."

Accepted Answer

"y² = $-$64x

focus : ($-$16, 0)

y = mx + c is focal chord

$\Rightarrow$ c = 16 m ...........(1)

y = mx + c is tangent to (x + 10)² + y² = 4

$\Rightarrow$ y = m(x + 10) $\pm$ 2$\sqrt {1 + {m^2}}$

$\Rightarrow$ c = 10m $\pm$ 2$\sqrt {1 + {m^2}}$

$\Rightarrow$ 16m = 10m $\pm$ 2$\sqrt {1 + {m^2}}$

$\Rightarrow$ 6m = 2$\sqrt {1 + {m^2}}$ (m > 0)

$\Rightarrow$ 9m² = 1 + m²

$\Rightarrow$ m = ${1 \over {2\sqrt 2 }}$ & c = ${8 \over {\sqrt 2 }}$

$4\sqrt 2 (m + c) = 4\sqrt 2 \left( {{{17} \over {2\sqrt 2 }}} \right)$ = 34"

Let y = mx + c, m > 0 be the focal chord of y² = $-$ 64x, which is tangent to (x + 10)² + y² = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________.

Solution

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Let y = mx + c, m > 0 be the focal chord of y2 = $-$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________.

Solution

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Let y = mx + c, m > 0 be the focal chord of y² = $-$ 64x, which is tangent to (x + 10)² + y² = 4. Then, the value of 4$\sqrt 2$ (m + c) is equal to _____________.