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If PSQ is a focal chord of the ellipse 1...

If PSQ is a focal chord of the ellipse `16x^(2)+25y^(2)=400` such that SP=16, then the length SQ is

A

`2/9`

B

`4/9`

C

`8/9`

D

`16/9`

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The correct Answer is:
To solve the problem step by step, we will follow the mathematical reasoning provided in the video transcript. ### Step 1: Write the equation of the ellipse in standard form. The given equation of the ellipse is: \[ 16x^2 + 25y^2 = 400 \] To convert it to standard form, we divide the entire equation by 400: \[ \frac{16x^2}{400} + \frac{25y^2}{400} = 1 \] This simplifies to: \[ \frac{x^2}{25} + \frac{y^2}{16} = 1 \] From this, we can identify \( a^2 = 25 \) and \( b^2 = 16 \). Therefore, we have: \[ a = 5 \quad \text{and} \quad b = 4 \] ### Step 2: Identify the length of the semi-latus rectum (LR). The formula for the length of the latus rectum of an ellipse is given by: \[ \text{Length of LR} = \frac{2b^2}{a} \] Substituting the values of \( b \) and \( a \): \[ \text{Length of LR} = \frac{2 \times 16}{5} = \frac{32}{5} \] ### Step 3: Use the harmonic mean (HM) of the segments of the focal chord. We know that the semi-latus rectum is the harmonic mean of the segments of the focal chord \( SP \) and \( SQ \). The formula for the harmonic mean (HM) of two numbers \( SP \) and \( SQ \) is: \[ \text{HM} = \frac{2 \cdot SP \cdot SQ}{SP + SQ} \] Given \( SP = 16 \), we can set up the equation: \[ \frac{2 \cdot 16 \cdot SQ}{16 + SQ} = \frac{32}{5} \] ### Step 4: Solve for \( SQ \). Cross-multiplying gives: \[ 2 \cdot 16 \cdot SQ = \frac{32}{5} \cdot (16 + SQ) \] This simplifies to: \[ 32SQ = \frac{32}{5}(16 + SQ) \] Multiplying both sides by 5 to eliminate the fraction: \[ 160SQ = 32(16 + SQ) \] Expanding the right side: \[ 160SQ = 512 + 32SQ \] Now, rearranging the equation: \[ 160SQ - 32SQ = 512 \] \[ 128SQ = 512 \] \[ SQ = \frac{512}{128} = 4 \] ### Final Answer: The length of \( SQ \) is \( 4 \).
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